When I first started on the Internet back in 1995 I think WS_FTP was the FTP program I first used. It always worked well until I couldn't get it to work properly for some reason. It was one issue after another. I don't even recall what the issue was but I found the "free" WinSPC so I was happy. I don't do a lot of FTPing anymore now that I switched from Windows/ColdFusion servers to Linux servers. Seems like it has been an eternity ago!
Having migrated a large site using FileZilla I've found that several files were ommitted from the download from the old server or upload to the new server, in particular image files.
Luckilly I also had a recent backup of the website on my PC to refer back to to recover the missing image files from.
If you don't mind paying for something, use FlashFXP. Only $29,95 for a lifetime license and you can use that on 4 pc's.
It's very flexible, I'm already using that for more then 15 years. Never had an issue with it, good support and updates if needed.
I use WinSCP for general SecureFTP from my PC, like uploading Xenforo addons, but for moving an entire forum/site that wouldn't be realistic.
If you're able to obtain some sort of command-line access, it would be worthwhile to learn to use a few command-line steps to archive your site (including hidden/all files) to a tar.gz or zip; and then moving it to a new server/host and extracting it there.
I use a server control panel, and it's respective backup feature, but also have two other backup types/destinations. All three run a separate times every 24 hours. For me, when the unexpected happens, the path to recovery hasn't always turned out as planned.
If you're looking for a paid software, and definitely better than CuteFTP, and looking for tons of features like file watchers, terminals, terminal scripts, folder sync, and navigation sync, and lots of automation tools blah blah, etc, there is no doubt SmartFTP is killing it.
But if you're looking for something free, I recommend WinSCP